Compactness property of Lie polynomials in the creation and annihilation operators of the q -oscillator
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Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator Rafael Reno S. Cantuba1 Received: 16 October 2019 / Revised: 5 June 2020 / Accepted: 18 June 2020 © Springer Nature B.V. 2020
Abstract Given a real number q such that 0 < q < 1, the natural setting for the mathematics of a q-oscillator is an infinite-dimensional, separable Hilbert space that is said to provide an interpolation between the Bargmann–Segal space of holomorphic functions and the Hardy–Lebesgue space of analytic functions. The traditional basis states are interrelated by the creation and annihilation operators. Since the commutation relation is q-deformed, the commutator algebra for the creation and annihilation operators is not a low-dimensional Lie algebra like that for the canonical commutation relation. In this study, a characterization of the elements of the said commutator algebra is obtained using spectral properties of the creation and annihilation operators as these faithfully represent the generators of a q-deformed Heisenberg algebra. The derived algebra of the commutator algebra is precisely the set of all compact operators, and the resulting Calkin algebra is algebraically isomorphic to the complex algebra of Laurent polynomials in one indeterminate. As for any operator that is not in the commutator algebra, the action of such an operator on an arbitrary basis state can be approximated by a Lie series of elements from the commutator algebra. Keywords q-oscillator · q-deformed Heisenberg algebra · Creation operator · Annihilation operator · Commutation relation · Commutator algebra · Lie algebra · Unilateral weighted shift · Calkin algebra · Compact operator · Fredholm operator · Commutator · Lie polynomial · Laurent polynomial Mathematics Subject Classification 17B10 · 17B15 · 17B81 · 17B37 · 47B07 · 47B15 · 47B32 · 47B37 · 47B47 · 47L30 · 47L90 · 46H70 · 81R10 · 81R50 · 81R99
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Rafael Reno S. Cantuba [email protected] Mathematics and Statistics Department, De La Salle University, Manila, 2401 Taft Ave., Malate, Manila 1004 Metro Manila, Philippines
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R. R. S. Cantuba
1 Introduction Consider a real number q ∈ ]0, 1[, and let H = Hq denote an infinite-dimensional separable Hilbert space over the complex field C. Fix a complete orthonormal basis for some dense subset of H, which consists of the usual ket vectors |n, where n ranges over all nonnegative integers. Such basis elements are generated from the basis element |0 by the creation operator a + . More precisely, a + |n = cn |n + 1 for some nonzero scalar cn for any n. The operator a + has an adjoint a called the annihilation operator, and these two operators satisfy the q-deformed commutation relation aa + − qa + a = 1.
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The Hilbert space H is the natural setting for the mathematics of the q-deformed quantum harmonic oscillator or simply q-oscillator, about which the literature is vast. We mention but only a few seminal and crucial ones. One of the main theoretical developments is the perspective that H provi
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